Bifurcation properties of Dicke Hamiltonians

Abstract

A variation in the coupled order parameter treatment of Dicke Hamiltonians in thermodynamic equilibrium is presented. The Hamiltonian is linearized by introducing disposable c‐number parameters. These parameters are chosen to minimize the resulting free energy. This requirement leads to a system of coupled nonlinear equations whose bifurcation properties are studied. The solution branches are labelled by the inertia of the free energy stability matrix. We prove that the parameters on the solution branch which provide the global minimum free energy also produce a linearized Hamiltonian thermodynamically equivalent to the original Hamiltonian provided only a finite number of field modes are present. This method is used to discuss the bifurcation and stability properties of the Dicke Hamiltonian with A² and counterrotating terms. We also discuss why the phase transition disappears in the presence of external currents or fields. We show how an internal gauge destroying mechanism may lead to the persistence of the phase transition even in the presence of external coupling. The method is used to discuss the phase transitions and multiplicity of ordered state phases in multilevel molecular systems. We also present a simple method for determining whether an external source will or will not destroy a second order phase transition and discuss the conditions under which such models may exhibit first order phase transitions.